Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, 269-278
© 2013 Journal of Mechanical Engineering. All rights reserved. DOI:10.5545/sv-jme.2012.478
(1.01) Original Scientific Paper
Received for review: 2012-03-23
Received revised form: 2013-01-18
Accepted for publication: 2013-02-12
Measurement of Involute Profile with Small Clamping
Eccentricity in a Gear Measuring Center
Lianfu, H. – Wenyan, T. – Jun, W. – Changfeng, F.
Han Lianfu1,2,* – Tang Wenyan1 – Wang Jun1 – Fu Changfeng2
1 Harbin
Institute of Technology, Institute of Precision Instrument, China
Petroleum University, School of Electronic Science, China
2 Northeast
This paper studies the influence of small clamping eccentricity on the measurement and evaluation of involute profile, analyzes the cause
of clamping eccentric error, establishes a clamping eccentric error compensation model, and introduces a computing method of clamping
eccentric parameters. Experimental testing of the computing method of clamping eccentric parameters proves that the computing method of
clamping eccentric parameters is correct and feasible when the clamping eccentric value is within three quarters of the measurement range
of the probe. The results of the clamping eccentricity error compensation experiment show that the clamping eccentric error compensation
model can compensate measurement results to the results attained under operating conditions, and achieve precision measurement of
involute profiles. The highest allowed clamping eccentric value can be extended to three quarters of the measurement range of the probe
by the clamping eccentricity error compensation model proposed in this paper, thereby saving time otherwise spent adjusting for clamping
eccentricity.
Keywords: gear measuring center, clamping eccentricity, involute profile, measurement, evaluation
0 INTRODUCTION
Involute gears are widely used in mechanical
manufacturing, and the quality of the involute
profile affects gear vibration and noise [1] and [2].
Measurement of involute profile is difficult, and
the involute profile is usually measured by a gear
measuring center [3] and [4]. Their measuring
principle is the Electronic-generating method, which
demands that the rotation of the center axis of the gear
measuring center should coincide with the geometric
center line of the gear [5] to [7]. It is easy to meet the
clamping demand for middle and small size external
gears with a core shaft and special clamp. But internal
gears cannot be clamped on the measurement platform
with a core shaft and special clamp because there is
no position to clamp internal gears. Large gears also
cannot be clamped on the measurement platform
because they are too large to find a proper core shaft
[7]. Large and internal gears are clamped on the
measurement platform of the gear measuring center
using a number of magnets. In the clamping process of
large gears and internal gears, a manual method with
a dial indicator is involved to assist in the clamping.
Therefore, the geometric center line of the gear often
parallel deviates from the rotation center axis of the
gear measuring center due to the limitations of manual
micro-positioning ability and the accuracy of the
auxiliary clamping instrument, resulting in clamping
eccentricity, where the error caused by this is called
the clamping eccentric error.
The clamping eccentric error inevitably
reduces the measurement accuracy of gear measuring
center. To decrease the influence of the clamping
eccentricity on the measurement of the involute
profile, the value of the clamping eccentricity is
adjusted to be as small as possible, but this takes a
lot of time and the clamping eccentricity still exists
[8]. Therefore, accurate measurement of the involute
profile with clamping eccentricity is a real problem.
A method of compensating for clamping eccentricity
error was proposed by Yang, and the method achieves
clamping eccentric error compensation by correcting
the transformation of the workpiece coordinate system
and the instrument coordinate system. However, this
method needs specially designed measurement nets
to obtain the proper clamping eccentricity parameters
[9]. Shi proposes a method to compensate for clamping
eccentricity [10], which measures the net points on the
involute profile and uses the least square method to
estimate the clamping eccentricity parameters, but
the method needs to include an extra measurement
trajectory for the computation of eccentricity
parameters. At present, the gear measuring center made
by Klingelnberg allows the measurement of involute
profile with clamping eccentricity. The allowed
maximum eccentricity value is 2 mm and the eccentric
parameters are obtained by measurement of the core
shaft. However, its disadvantage is its inaccessibility
to internal and large gears [11].
This paper proposes a measurement method
for involute profiles with small clamping eccentricity
for internal gears and large gears based on a gear
measuring center. The method not only allows
precision measurement of involute profiles without
changing the measurement trajectory, but also reduces
the time needed to adjust the clamping eccentricity.
*Corr. Author’s Address: Harbin Institute of Technology, 92 West Dazhi Street 150001, Harbin , China, lianfuhan@hit.edu.cn
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Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, 269-278
1 THE INFLUENCE OF CLAMPING ECCENTICITY ON
MEASUREMENT AND EVALUATION OF INVOLUTE PROFILE
To measure the involute profile with a small
clamping eccentricity, the influence of the clamping
eccentricity on the measurement and evaluation of
the involute profile should be considered. When
there is no clamping eccentricity, the measurement
trajectory of the probe relative to the gear is an
involute curve attained through the linkage of the
X-axis and main shaft. If the X-axis and main shaft
still maintain their linkage in the default way in the
clamping eccentric state, the profiles deviate from
their default measurement position by the existence of
the clamping eccentricity. So, any point P0(x0, y0, z0)
attained by generating the measured values of involute
profiles deviates from its default measurement
position, as shown in Fig. 1a. The measured points are
in the same plane, because the Z-axis remains static
when the involute profile is measured. Although there
is clamping eccentricity, the measured curve is part
of the involute curve to be measured and the point
P0(x0, y0, z0) is still in the curve to be measured. The
point P0(x0, y0, z0) may delay or advance its actual
position, and it may appear at position P1-point or
P2-point, both of which are on the measured involute
profile, as shown in Fig. 1b.
It is important to study the usage of the above
rule in measuring the involute profile with clamping
eccentricity without changing the measurement
trajectory. It is assumed that involute profiles are
by default in their measurement position during the
process of measurement of involute profiles in the
clamping eccentric measurement state, so the probe
may impact the gear if the clamping eccentricity is
too large. The measurement range of the probe is L,
so the maximum allowed deviation Δdmax between the
rotation center of the instrument and the geometric
center of the gear is expressed as:
The output and input of the probe have a strictly
linear relationship within three quarters of the
measurement range of the probe, so the appropriate
maximum value of the clamping eccentric value Δe is
given by Eq. (2).
b)
Fig. 1. The influence of clamping eccentricity on measurement
and evaluation of the involute profile; a) gear in the clamping
eccentricity condition, and b) the schematic diagram of a section
270
Δe = 0.75 L.(2)
For all kinds of gears, except for micro gears,
the relation between d1 (the distance from the
starting point of evaluation to the starting point of
measurement) and d2 (the distance from the end point
of evaluation to the end point of measurement) can be
expressed as:
a)
Δdmax = L.(1)
 d1 ≈ 0.5( LAE − Lα ) >> ∆e = 0.75 L
, (3)

d 2 ≈ 0.5( LAE − Lα ) >> ∆e = 0.75 L
where, LAE is the abbreviation of active length and Lα is
the abbreviation of the profile evaluation range [12].
According to Eq.(3), the length of the profile
that is not evaluated is much larger than the clamping
eccentric value. The clamping eccentricity only
affects the lengths of d1 and d2, and the evaluation
length is not affected when the clamping eccentricity
is within 0.75 L. So, the part of the curve involved
in the evaluation can be measured without adjusting
the measurement trajectory, but the profiles deviate
from their default measurement position due to the
existence of the clamping eccentricity. As shown in
Fig. 1b, the measurement values that stand for the P0point in the operation condition stand for the P1-point
in the clamping eccentricity condition, an error of
PM
is caused during the total deviation computation
1
of the tooth profile. The clamping eccentricity error
should be compensated for in the evaluation process of
involute profiles, although the measurement trajectory
of involute profile does not need to be changed when
the clamping eccentric value is 0.75 L.
Lianfu, H. – Wenyan, T. – Jun, W. – Changfeng, F.
Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, 269-278
2 ˝THE CLAMPING ECCENTRIC ERROR
COMPENSATION MODEL
2.1 Cause of the Clamping Eccentric Error
To establish the clamping eccentric error compensation
model, the cause of the clamping eccentric error
should be studied. The coordinates of involute
profiles in the workpiece coordinate system should
be transformed to the instrument coordinate system.
While evaluating involute profiles, the measured
values in the instrument coordinate system should be
transformed into the workpiece coordinate system.
Therefore, it is necessary to study the transformational
relationship between these two coordinate systems.
The workpiece coordinate system is established with
the geometrical center of the gear as the origin, and
the X, Y and Z directions of the instrument coordinate
system as X0, Y0 and Z0 directions of the workpiece
coordinate system, as shown in Fig. 2. The geometric
center line of the gear parallel deviates from the
rotation center axis of gear measuring center in the
clamping eccentric state and the gear is clamped onto
the platform of a gear measuring center, the origin
of workpiece coordinate system in the instrument
coordinate system keeps changing and it is denoted as
O0i.
a)
 xi   x0ii   0   x0ii 
       
 yi  =  y0ii  +  0  =  y0ii  + α , (4)
 zi   z   b2   z 
 0ii 
 0ii 
where b is the face width, and α is the translation
vector of the Z direction between the workpiece
coordinate system and the instrument coordinate
system in the ideal state.
As shown in Fig. 2b, the transformational
relationship of the coordinates in the clamping
eccentric state can be expressed as:
 xi   x0ii   0   mi   x0ii 
         
 yi  =  y0ii  +  0  +  ni  =  y0ii  + α + β i , (5)
 zi   z   b2   0   z 
 0ii 
 0ii 
where βi is the clamping eccentricity transformational
vector between the workpiece coordinate system and
the instrument coordinate system, and (mi, ni, b/2) is
the origin of the workpiece coordinate system in the
instrument coordinate system.
When the clamping eccentric value is small,
βi ≈ 0 and Eq. (5) is approximately valid if βi is
neglected. Under this condition, the influence of
neglecting βi on the transformational accuracy of
the coordinates is tiny and the clamping eccentric
error can be neglected. When the clamping eccentric
value is large, βi > 0 and Eq. (5) is not valid if βi is
neglected. The influence of neglecting βi on the
transformational accuracy of the coordinates is
large, so it would result in a large transformational
error of the coordinates. Under this condition, the
clamping eccentric error should not be neglected,
but in fact the clamping eccentricity transformational
vector βi is still not considered, so the clamping
eccentric error appears. From the preceding analysis
it is clear that the clamping eccentric error is caused
by the transformation error between the workpiece
coordinate system and the instrument coordinate
system due to ignoring the clamping eccentricity
transformational vector βi.
2.2 Establishment of the Clamping Eccentric Error
Compensation Model
b)
Fig. 2. Construction of the workpiece coordinate system; a) in the
ideal state, and b) in the clamping eccentric state
As shown in Fig. 2a, the transformational
relationship of the coordinates in the ideal state is
expressed as:
According to Eq. (5), βi is determined by the origin
O0i(mi, ni) of the workpiece coordinate system in the
instrument coordinate system. Because the gear is
clamped on the measurement platform of the gear
measuring center, the distance between the origin of
the workpiece coordinate system and the origin of
Measurement of Involute Profile with Small Clamping Eccentricity in a Gear Measuring Center
271
Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, 269-278
the instrument coordinate system is a constant. When
the gear revolves 2π around the rotation center axis
of gear measuring center, the geometric center of
the gear moves in circle O, as shown in Fig. 3. It is
assumed that the origin of workpiece coordinate
system at a certain measurement time is at the position
point O00, the clamping eccentric value that the initial
position of the gear geometrical center axis deviates
from the rotation center axis is e0 = e, and the polar
angle is ϕ0 and the reading of main shaft is θ0. After
main shaft rotates θi, the origin of the workpiece
coordinate system revolves to point O0i. As shown in
Fig. 3, the eccentric value ei and the polar angle ϕi at
this measurement moment can be obtained as:
ei = e
, (6)

ϕi = ϕ0 + θi
where θi is the angle that the main shaft rotates to the
initial angle θ0.
The origin of the workpiece coordinate system in
the instrument coordinate system is point O0i(mi, ni),
and the equations of the orgin are as follows:
mi = e cos ϕi = e cos(ϕ0 + θi )
. (7)

 ni = e sin ϕi = e sin(ϕ0 + θi )
the ith workpiece
coordinate system
the i-position
polar angle
clamping eccentric
Y
value
between the instrument coordinate system and the
i-position of workpiece coordinate system is expressed
as Eq.(8):
The measured values of involute profiles should
be transferred into the same workpiece coordinate
system in the evaluation. In this paper, the measured
values are transfered into the initial workpiece
coordinate system. As shown in Fig. 3, the coordinate
transformational relationship between the initial
workpiece coordinate system and the i-position of
workpiece coordinate system can be expressed as:
e
O
main shaft
rotating angle
 x0ii 
 
 y0ii  (9)
z ,
 0ii 
 1 
where, M is the coordinate transformational matrix
between the initial workpiece coordinate system and
the i-position of the workpiece coordinate system and
it is expressed as:
1
0
M =
0

0
0
1
0
0
0 −e[cos φ0 − cos(φ0 + θi )]
0 e[sin(φ0 + θi ) − sin φ0 ] 
,(10)

1
0

0
1

The clamping eccentric error compensation
model is established according to Eqs. (8) and (9), and
the expression of the model is given by Eq. (11).
X
the initial workpiece
coordinate system
initial polar angle
Fig. 3. Running track of the gear’s geometrical center
The No.i workpiece coordinate system
When the origin of the workpiece coordinate
system is at the position point O0i, the workpiece
coordinate system is denoted as the ith workpiece
coordinate system. For any point Pi(xi, yi, zi, θi) in the
instrument coordinate system, according to Eqs. (2)
and (7), the coordinate transformational relationship
272
 x00i 
 
 y00i 
z  = M
 00i 
 1 
Y0i
O0i Y00 X 0i
φ
θi i
X 00
φ0 O00
 xi   x0ii 
e cos(ϕ0 + θi ) 
   


 yi  =  y0ii  + α +  e sinn(ϕ0 + θi )  . (8)
 zi   z 


0
 0ii 
 x00i 
 xi − e cos(φ0 + θi ) 
 
 y − e sin(φ + θ ) 
y
 00i 
0
i 
 i
=
. (11)
M
b
z 


−
z
i
2
00i 



1
 1 


Once the coordinate transformation without error
between the initial workpiece coordinate system and
the instrument coordinate system is realized by the
clamping eccentric error compensation model, then
the involute profiles can be evaluated according to the
evaluation standard. According to Eq. (11), there are
only two unknown parameters (clamping eccentric
value e and polar angle ϕ0) in the model.
Lianfu, H. – Wenyan, T. – Jun, W. – Changfeng, F.
Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, 269-278
2.3 Calculation of Clamping Eccentric Parameters e
and ϕ0
This part discusses the calculation of the clamping
eccentric parameters according to the measuring
points of the tooth profile. As shown in Fig. 4a, the
probe is adjusted to the base circle in the X direction.
The proportional relationship between the rotation
angle of the main shaft and the moving distance of
the Y-guideway is determined by the parameters
of the gear to be measured. The geometrical center
O00 (m0, n0) of the gear at a certain measurement
moment is defined as the origin of the initial
workpiece coordinate system, as shown in Fig. 4b.
As shown in Fig. 4b, the involute curve P0Pm is
the involute profile to be measured, and O00 (m0, n0)
is the center of the base circle and rb is the radius.
The radius vector rkm at the point Pm (xm, ym) of the
involute profile is expressed as:
rkm = ( xm − m0 ) 2 + ( ym − n0 ) 2 . (13)
As shown in Fig. 4b, involute curve P0Pm is the
involute profile to be measured, and O00 (m0, n0) is the
center of the base circle and rb is the radius.
According to the characteristics of the involute,
the expression for generating angle θ km at the point
Pm (xm, ym) of the involute profile is as follows:
θ km = tan α km − α km , (14)
Y
Base circle
Probe of gear
measuring center
X
Geometric circle of gear
where, α km is the pressure angle at the point
Pm (xm, ym) of the involute profile and can be expressed
as:
rb

 α km = arccos r
km

. (15)

2
rkm − rb2

 tan α km =
rb

Rotation circle of gear
measuring center
a)
As shown in Fig. 4b, βm is expressed as Eq. (16):
β m = ∠DO00 P0 − ∠DO00 Pm =
y −n
y −n
= arctan 0 0 − arctan m 0 . (16)
x0 − m 0
xm − m0
As shown in Fig. 4b, βm can also be expressed as
Eq. (17).
β m = θ km − θ k0 . (17)
Let
b)
Fig. 4. Determination of circle center using the involute curve;
a) measurement of involute profiles, and b) example of an involute
curve
The clamping eccentric angle ϕ0 and the clamping
eccentric value e can be expressed as:
φ0 = ∠O00OX
, (12)

 e = O00O
f m = θ km − θ k0 − ∠DO00 P0 − ∠DO00 Pm . (18)
Eq. (19) should hold in theory, but it does not hold
for the manufacturing error and measurement error.
f m = 0. (19)
According to the principle of the least square
method, the following model for the identification of
clamping eccentric parameters is established:
N

2
 F (m0 , n0 , rb ) = min ∑ f m
. (20)

m =1

>
0
s
.
t
r
b

Measurement of Involute Profile with Small Clamping Eccentricity in a Gear Measuring Center
273
Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, 269-278
Eq. (20) is a nonlinear least square problem and it
can be solved by recursive methods [13]. In theory, as
long as the measurement values of one tooth profile are
obtained, m0 and n0 can be estimated. But it may result
in a large deviation between the obtained eccentricity
parameters and the true value. To reduce such a large
deviation, the measurement values of teeth evenly
distributed in the gear are adopted. The circle center is
obtained from each tooth is O0i (m0i, n0i). The optimum
identification model of the circle center is established
as follows:
is determined by the parameters of the gear to be
measured. The involute profile with small clamping
eccentricity is compensated such that it is in the ideal
state using the clamping eccentric error compensation
model. Involute profiles are evaluated according to
ISO1328-1 [12].
z0
d (m0 , n0 ) = min ∑ [(m0i − m0 ) 2 + (n0i − n0 ) 2 ], (21)
i =1
where, z0 is the measured number of teeth. The
binary function d (m0, n0) has its minimum value
at the inflexion, so we take the partial derivative
of d (m0, n0) with respect to m0 and n0 to find the
inflexion.
 ∂d (m0 , n0 )
=0
 ∂m

0
. (22)

 ∂d (m0 , n0 ) = 0
 ∂n0
The inflexion (m0, n0),which is also the optimized
circle center, can be obtained using Eq. (22). The
clamping eccentric value e and the eccentric angle ϕ0
expressed as:
ϕ0 = arctan(n0 / m0 )
. (23)

2
2
 e = m0 + n0
Fig. 5. The experimental equipment
3.1 Experiment on the Calculation Method Of Clamping
Eccentric Parameters
The dial indicator is used to monitor the geometrical
center position of the internal gear and to control the
clamping eccentric value within 0.75L. The points of
the addendum circle are measured every 0.5° by the
gear measuring center. As shown in Fig. 6, pi (xi, yi)
is one measurement point in the addendum circle, the
circle center o2 (aˆ , bˆ) is obtained using the measured
values of the points in the addendum circle using the
Limacon Model [14].
3 EXPERIMENTS
To verify the correctness of the calculation method
for clamping eccentricity parameters and the validity
of the clamping eccentricity error compensation
model, the following experiments were carried out
with the internal gear on a L45 type gear measuring
center. The measurement range of the probe in the
experiment is 0.70 mm and the parameters of the
internal gear are as follows: modulus is 3.25; tooth
number is 60; modification coefficient is 0; face width
is 30 mm; pressure angle is 20°; spiral angle is 0°.
Fig. 5 shows the measurement equipment. The gear
is fixed on the platform with six magnets, the probe
is adjusted to the base circle in the X direction. The
proportional relationship between the rotation angle of
main shaft and the moving distance of the Y-guideway
274

a =


b =

1 N −1
∑ xi
N i =0
1 N −1
∑ yi
N i =0
. (24)
The involute profiles of the gear are measured
under the same clamping eccentric condition,
and the center of base circle is obtained using the
Involute Model proposed in this paper. To decrease
contingency, a series of experiments were carried out
under different clamping eccentric value conditions.
3.2 The Clamping Eccentric Error Compensation
Experiment
Using the Involute Model proposed in this paper,
the clamping eccentric value is 164.3 mm and the
clamping eccentric angle is 43.62°. The clamping
eccentric error compensation model is applied in the
evaluation of the involute profiles.
Lianfu, H. – Wenyan, T. – Jun, W. – Changfeng, F.
Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, 269-278
y
p
i
(
xi yi
,
)
measurement curve
optimize fitted curve
ri
θi
O2(a,b)
o1
x
r0
Fig. 6. Limacon Model
To decrease the influence of chance in the error
compensation experiment, the position of the gear
is adjusted and the involute profiles are measured
20 times for each clamping eccentric value. Then
the preceding operation is repeated with a different
clamping eccentric value.
3.3 The Experiment of Saving Time in Using Adjusted
Clamping Eccentricity
To compare the saving time in using adjusted clamping
eccentricity, a number of experiments were carried out
with a series of gears whose tip diameter is less than
3000 mm. To ensure the accuracy of the measurement,
the clamping eccentric value is required to be as small
as possible. For example, for a gear whose tip diameter
is no more than 450 mm, the clamping eccentric value
is usually required to be less than 2 μm. The highest
allowed clamping eccentric value can extend to 0.75
L using the clamping eccentricity error compensation
model proposed in this paper.
4 RESULTS AND DISCUSSION
4.1 Results and Discussion of the Calculation Method of
Clamping Eccentric Parameters
In theory, the circle center of the addendum circle
should coincide with the circle center of the base
circle, and the clamping eccentric value and the
clamping eccentric angle can be attained by the
circle center according to Eq. (23). Table 1 shows a
comparison of the results of the experiment on the
calculation method of clamping eccentric parameters.
As shown in Table 1, both the eccentric values
attained by Limacon Model and those attained by
the Involute Model basically tally with those using
the Indicator. The maximum deviation of clamping
eccentric values is 1.9 mm, which can be ignored
compared to the maximum allowed clamping
eccentric value of 0.75 L. The maximal deviation
of the eccentric angle between two models is 0.09°,
which can also be ignored. This shows that the
eccentricity parameter estimation method raised in
this paper is correct and feasible.
4.2 Results and Discussion of the Clamping Eccentric
Error Compensation Experiment
The measurement results before compensation are
shown in Fig. 7a, then the clamping eccentric error
compensation model is applied to the evaluation of
the involute profiles and the deviation diagrams are
shown in Fig. 7b.
The measurement results are shown in Table 2.
There are eight deviation curves of involute
profiles in every deviation diagram in Fig. 7, and the
deviation distance from the vertical lines in every
curve stands for the deviation of the involute profiles.
The deviation curves of involute profiles should be
straight lines if there are no deviation in the profiles,
and there is 1 mm per unit of profile deviation.. To
display the deviation clearly, the deviations are
enlarged 1000 times in Fig. 7. The numbers “1, 16, 31,
46” mean the numbers of the measured tooth, and the
numbers on the left stand for the number of left tooth
profiles and those on the right stand for the number
of right tooth profiles. Fig. 7a and c are obtained
under the same measurement condition for the same
four involute profiles. However, for the clamping
eccentricity, the inclined degree of the deviation
curves is obviously different, which indicates that
±fHα (profile slope deviation) increases greatly. The
curve shape of the deviation curves is also different,
which means that ffα (profile form deviation) and Fα
(total profile deviation) are affected by the change in
Table 1. Estimated clamping eccentric parameters of different models
Limacon
Model
Involute
Model
Dial Indicator
e [μm]
Φ0 [°]
e [μm]
Φ0 [°]
e [μm]
29.3
3.30
29.1
3.28
30
69.1
355.27
68.7
355.34
70
100.4
42.45
98.5
42.36
100
164.4
43.71
164.3
43.62
160
228.5
9.29
227.2
9.33
230
299.8
186.62
298.7
186.71
300
Measurement of Involute Profile with Small Clamping Eccentricity in a Gear Measuring Center
357.2
190.78
356.1
190.74
360
491.7
6.78
490.3
6.73
490
275
Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, 269-278
Table 2. Measurement results of involute profiles under different conditions
Under operating conditions
Before compensation
After compensation
Fα [μm]
8.0
35.7
8.3
Grade
5
9
5
the evaluation position resulting from the clamping
eccentricity. As shown in Table 2, due to clamping
eccentricity, the 5th grade Fα is incorrectly judged
as grade 9, the 6th grade Ffα is incorrectly judged as
grade 10, and the 4th grade ±fHα as grade 9. Therefore,
the measurement results are seriously affected. So,
the measurement of tooth profile is seriously affected
under conditiond of clamping eccentricity if the
clamping eccentricity compensation is not taken into
account. After compensation, the curves in Fig. 7b
mostly accord with those in Fig. 7c in the evaluation
range. According to Table 2, the measurement results
after compensation are close to those under operating
conditions and do not influence the evaluation grade.
ffα [μm]
7.9
37.2
7.6
Grade
6
10
6
± fHα [μm]
-3.2
-17.8
-3.1
Grade
4
9
4
The average measurement results of the
involute profiles before and after compensation with
different clamping eccentric values are shown in Fig.
8. As shown in Fig. 8, as the clamping eccentric value
grows, the measurement errors of Fα, Ffα and ±fHα all
increase. The maximum measurement errors of Fα, Ffα
and ±fHα are, respectively, 31.6, 29.5 and 16.6 μm, so
the measurement results seriously deviate from the
true results under operating conditions.
a)
a)
b)
b)
c)
Fig. 7. Comparison of measurement results; a) the measurement
result before compensation, b) the measurement result after
compensation, c) the measurement result under operating
conditions
276
c)
Fig. 8. Comparison of profile deviation before and after
compensation; a) comparison of total profile deviation,
b) comparison of profile form deviation, and c) comparison of
profile slope deviation
Lianfu, H. – Wenyan, T. – Jun, W. – Changfeng, F.
Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, 269-278
After compensation, the following measurement
results are attained.
• The average value of Fα is 8.1 mm, which
approximates to the true value 8.0 mm. Fα is the
5th grade of accuracy class, which is also the
same as the true grade.
• The average value of Ffα is 7.9 mm, which
approximates to the true value. Ffα is the 6th
grade, which is the same as the true grade.
• The average value of ±fHα is –3.3 mm, which
approximates to the true value –3.4 mm. ±fHα is
the 4th grade of accuracy class, which is also the
same as the true grade.
From the results of the preceding experiments,
the measurement results after compensation
basically tally with those obtained under operating
conditions. Therefore, the clamping eccentricity error
compensation model is correct.
4.3 Results and Discussion of the saving time in adjusting
the clamping eccentricity
The highest allowed clamping eccentric value can
extend to 0.75 L using the clamping eccentricity error
compensation model proposed in this paper. The time
saved adjusting the clamping eccentricity is shown in
Fig. 9.
5 CONCLUSIONS
This paper draws the following conclusions.
• The measurement trajectory of involute profiles
does not need to be changed if the clamping
eccentric value is within 0.75 L, but the
profiles should be compensated to the default
measurement position in the evaluation.
• The clamping eccentric error is caused by a
coordinate transformation error between the
workpiece coordinate system and the instrument
coordinate system when the clamping eccentricity
transformational vector βi is ignored.
• The eccentricity parameters obtained using the
involute Model raised in this paper are consistent
with those obtained using the Limacon model.
Therefore, the eccentricity parameters estimation
method proposed in this paper is correct and
feasible.
• The clamping eccentric error compensation
model compensates evaluation results to the
state without eccentricity, allowing precision
measurement of involute profile.
• The highest allowed clamping eccentric value
can extend to 0.75 L, which saves a lot of
time otherwise spent in adjusting clamping
eccentricity.
6 ACKNOWLEDGEMENT
This work was supported by the Technology Program
of the Educational Office of Hei Longjiang Province,
China (No.11553016).
7 REFERENCES
Fig. 9. Time saved adjusting clamping eccentricity
As shown in Fig.9, the time saved adjusting the
clamping eccentricity increases with the increase of the
addendum circle diameter. When the addendum circle
diameter of a gear is within 600 mm, the method can
save five minutes, on average. When the addendum
circle diameter of the gear is beyond 600 mm, the time
saved increases rapidly and the maximum time saved
is 187 minutes. All in all, the method of involute
profiles saves a lot of time that would otherwise be
spent adjusting clamping eccentricity.
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